Chance Constraints Integrated MPC Navigation in Uncertainty amongst Dynamic Obstacles: An overlap of Gaussians approach

In this paper, we formulate a novel trajectory optimization scheme that takes into consideration the state uncertainty of the robot and obstacle into its collision avoidance routine. The collision avoidance under uncertainty is modeled here as an overlap between two distributions that represent the state of the robot and obstacle respectively. We adopt the minmax procedure to characterize the area of overlap between two Gaussian distributions, and compare it with the method of Bhattacharyya distance. We provide closed form expressions that can characterize the overlap as a function of control. Our proposed algorithm can avoid overlapping uncertainty distributions in two possible ways. Firstly when a prescribed overlapping area that needs to be avoided is posed as a confidence contour lower bound, control commands are accordingly realized through a MPC framework such that these bounds are respected. Secondly in tight spaces control commands are computed such that the overlapping distribution respects a prescribed range of overlap characterized by lower and upper bounds of the confidence contours. We test our proposal with extensive set of simulations carried out under various constrained environmental configurations. We show usefulness of proposal under tight spaces where finding control maneuvers with minimal risk behavior becomes an inevitable task.